A continuous-time model for the additive white Gaussian noise (AWGN) channel in the presence of white (memoryless) phase noise is proposed and discussed. It is shown that for linear modulation the output of the baud-sampled filter matched to the shaping waveform represents a sufficient statistic. The analysis shows that the phase noise channel has the same information rate as an AWGN channel but with a penalty on the average signal-to-noise ratio, the amount of penalty depending on the phase noise statistic. © 2014 IEEE

Barletta, Luca

Kramer, Gerhard

arXiv

BARLETTA, LUCA

Archivio istituzionale della ricerca - Politecnico di Milano

On Continuous-Time White Phase Noise ChannelsLuca BarlettaInstitute for Advanced StudyTechnische Universita¨t Mu¨nchenD-85748 Garching, Germanyluca.barletta@tum.deGerhard KramerInstitute for Communications EngineeringTechnische Universita¨t Mu¨nchenD-80333 Munich, Germanygerhard.kramer@tum.deAbstract—A continuous-time model for the additive whiteGaussian noise (AWGN) channel in the presence of white(memoryless) phase noise is proposed and discussed. It is shownthat for linear modulation the output of the baud-sampled filtermatched to the shaping waveform represents a sufficient statistic.The analysis shows that the phase noise channel has the sameinformation rate as an AWGN channel but with a penalty on theaverage signal-to-noise ratio, the amount of penalty dependingon the phase noise statistic.I. INTRODUCTIONA major source of impairment in radio and optical channelsis multiplicative phase noise that arises due to instabilities oflocal oscillators used in up- and down-conversion. Estimationand compensation of phase noise is a classic problem incommunication systems theory. In fiber-optic communicationsystems, phase noise is present due to instabilities of laseroscillators [1] or due to cross-phase modulation in multi-channel systems [2].In the literature the phase noise process is often modeledas a discrete-time process that is output by a baud-sampledmatched filter [3], [4]. This approximate model is valid aslong as the variations of the phase noise process are small inone symbol period and the signal-to-noise ratio (SNR) is lowor moderate [5]. In general, and especially for phase noisecharacterized by large linewidths, the channel impaired byphase noise should be modeled and studied in continuous-time, in order to account for the effects of filtering prior tosampling [5]–[7].Analytical bounds on the capacity of phase noise channelsare given in [8], and for high SNR in [9]. Numerical bounds onthe information rate transferred through discrete-time Wienerphase noise channels are provided in [4], [10], [11], whilefor the continuous-time counterpart analytical lower boundsare given in [6], [7] and an analytical upper bound in [12].Numerical results for the achievable information rates in thecase of discrete-time phase noise with an arbitrary spectrumare given in [13], [14].Few papers in the literature deal with continuous-time phasenoise and, apart from trivial cases, the existence of a finite-dimensional sufficient statistic for these channels has not beeninvestigated yet. The aim of this paper is to attempt to modelthe continuous-time additive white Gaussian noise (AWGN)channel in the presence of white phase noise, and to find a(finite-dimensional) sufficient statistic.The motivation for studying white phase noise is that thereare problems for which the phase varies more rapidly thanthe bandwidth of the receiving device, e.g., a “square law”detector that necessarily has limited bandwidth. In such cases,the device will “lose” energy and we wish to model this effect.The paper is organized as follows. In Sec. II previous workson this topic are analyzed in detail. The system model isintroduced and discussed in Sec. III, while a sufficient statisticis derived in Sec. IV. Results are discussed in Sec. V, andconclusions are drawn in Sec.VI.II. PREVIOUS WORKThe output of a baud-sampled matched filter represents asufficient statistic for the continuous-time AWGN channel witha constant phase shift given by a random parameter (see, e.g.,[15, Sec. 4.4.1]).In the context of continuous-time Wiener phase noise chan-nels, the authors of [5] show through numerical simulationsthat a multi-sample receiver achieves significantly higher infor-mation rates than the baud-sampled matched filter, especiallyat high SNR. The dimensionality of the sufficient statistic forthis channel with memory has not been investigated.In the context of continuous-time memoryless phase noisechannels, the authors of [16, Eq. (28)] model the partiallycoherent channel asY (t) = X(t)ejΘ(t) +W (t), 0 ≤ t ≤ T (1)where W is a complex-valued AWGN process and Θ isa white phase noise process used to model the nonlineareffects of cross-phase modulation in multichannel fiber-opticcommunication systems. White means that Θ(t1) and Θ(t2)are uncorrelated random variables for any t1, t2 in the interval[0, T ] with t1 6= t2. If, in addition, Θ is stationary and Θ(t) isdistributed according to a wrapped Gaussian with zero meanand variance σ2, then the autocorrelation function RejΘ of themultiplicative disturbance ejΘ(t) isRejΘ = E[ejΘ(t)e−jΘ(t+τ)]= 1 τ = 0e−σ2 τ 6= 0 (2)= e−σ2+ limB→∞(1− e−σ2) · sinc(Bτ). (3)The power spectral density (PSD) Sy(f) of the received signalY (t) is given bySy(f) = F{Ry(τ)} = Sx(f) ? SejΘ(f) + Sw(f)= e−σ2Sx(f) + limB→∞Sx(f) ? (1− e−σ2) rect(f/B)B+ Sw(f), (4)where F{·} denotes the Fourier transform and ? the convo-lution operator. Equation (4) shows that the PSD of the inputsignal Sx(f) experiences a spectral loss [16], that is a gainequal to e−σ2 ≤ 1, and the remaining power proportional to(1− e−σ2) is spread onto an infinite bandwidth. Later in thispaper we will see that when using projection receivers thespectral loss is unavoidable.III. SYSTEM MODELConsider linear modulation for which we express the outputof the continuous-time white phase noise channel asY (t) = X(t)ejΘ(t) +W (t)=∑kAkg(t− kT )ejΘ(t) +W (t)=∑kAkgk(t)ejΘ(t) +W (t), (5)where {Ak} is a sequence of independent and identicallydistributed (iid) random variables, T is the symbol period, jis the imaginary unit, g(t) is the pulse shaping filter such that{gk(t)} is a set of orthonormal functions of L2(R) with theproperty ∫ ∞−∞gk(t)g?l (t) dt = 1 k = l0 k 6= l (6)where the superscript ? denotes complex conjugate, and W isa complex-valued circularly symmetric white Gaussian noise.Let{φnm(t), n = 0, 1, . . . ;m ∈ Z} (7)be a complete orthonormal basis of L2(R) withφnm(t) = φn(t−mT ) and such that∫ ∞−∞φnm(t)φ?n′m′(t) dt = 1 n = n′,m = m′0 otherwise, (8)and ∫ ∞−∞gk(t)φ?nm(t) dt = 1 n = 0,m = k0 otherwise. (9)The basis functions in (7) are double indexed: the index nmdenotes the n−th basis function φn(t) translated by mT intime, i.e., φn(t − mT ). The choice of using a double indexfor basis functions is because orthogonality is also obtained bytranslation, as shown in (8). Property (9) states that the pulseshaping function g(t) has a non-zero projection only in the firstdimension, i.e., for n = 0. This means that φ0m(t) = gm(t).Note that assuming property (9) comes without loss of general-ity, because we can always obtain all the other basis functionsφn(t) for n > 0 with an orthonormalization procedure.Example. A complete orthonormal basis of L2(R) is thetrigonometric system{φnm(t) = φn(t−mT ), n = 0, 1, · · · ;m ∈ Z} , (10)whereφn(t) =1√Texp(j2piTnt)(11)for 0 ≤ t < T and φn(t) is zero elsewhere. The square pulseshaping filterg0(t) = 1/√T 0 ≤ t < T0 otherwise(12)is such that (6) and (9) are satisfied.The phase noise process Θ is defined as a stationary randomprocess such thatE[〈gk(t)ejΘ(t), φnm(t)〉], µΘ · 1 n = 0,m = k0 otherwise,(13)E[〈gk(t)ejΘ(t), φnm(t)〉·〈gk′(s)e−jΘ(s), φn′m′(s)〉], |µΘ|2 · 1 n = n′ = 0,m = k,m′ = k′,0 otherwise, (14)where µΘ = E[ejΘ(t)]. Note that (13) and (14) giveVar[〈gk(t)ejΘ(t), φnm(t)〉]= 0 (15)for any choice of k, n, and m. In other words, we can viewthe projections〈gk(t)ejΘ(t), φnm(t)〉as constants.Defining the phase noise process through (13)-(15) is rea-sonable if we interpret the model (5) as the limit of successiverefinements of a related discrete-time model. For the discrete-time model, consider the finite interval [−S, S] and timeinstants ti = iS/l for i = −l, . . . , l − 1. The extension tothe whole real axis is straightforward by letting S → ∞.Since the continuous-time AWGN W (t) is defined only whenusing projection receivers, also for the phase noise process it isdesirable to preserve the statistical properties of its projectionsonto the basis functions, when passing from the discrete-timeto the continuous-time model. Mathematically speaking, it isuseful to define〈gk(t)ejΘ(t), φnm(t)〉, liml→∞12ll−1∑i=−lgk(ti)ejΘ(ti)φ?nm(ti),(16)where the equality represents some kind of convergence.The condition and the type of convergence is given by thefollowing Lemma.Lemma. If |gk(t)φnm(t)| is uniformly bounded in t, then theright-hand side of (16), that is the projection of the phasenoise process gk(t)ejΘ(t) onto the basis function φnm(t) ofL2([−S, S]), converges (almost surely) to the expected valueof the projection:〈gk(t)ejΘ(t), φnm(t)〉a.s.= µΘ∫ S−Sgk(t)φ?nm(t) dt, (17)for any value of k, n, and m.Proof: We show the convergence result for the real partof the projection; the procedure for the imaginary part isanalogous. Consider the independent random variablesZi+1 = <{gk(ti)φ?nm(ti)ejΘ(ti)}, i = 0, . . . , l−1, (18)where <{x} is the real part of complex number x, then Kol-mogorov’s criterion for the strong law of large numbers [17,Th. 1-3, pag. 238] for l→∞ reads as∞∑i=1Var [Zi]i2≤∞∑i=1E[<{gk(ti−1)φ?nm(ti−1)ejΘ(ti−1)}2]i2≤∞∑i=1|gk(ti−1)φnm(ti−1)|2i2≤ Kpi26<∞, (19)where (19) follows by the boundedness condition|gk(t)φnm(t)|2 ≤ K for any t ∈ [−S, S], with K < ∞. Thefiniteness of (19) implies the strong law of large numbers forindependent and non-identically distributed random variables:1ll∑i=1Zia.s.−→ liml→∞1ll∑i=1E [Zi]= <{µΘ∫ S0gk(t)φ?nm(t) dt}(20)where the last equality follows by the definitions of µΘ andof the Riemann integral.Proving the convergence in the time interval [−S, 0] isanalogous, and the two parts together prove the thesis.Result (17) is in accordance with the definitions (13)-(15).The condition of boundedness of the product of the basisfunctions required by the Lemma is mild and can be easilymet, therefore we assume it is met.Note that the proof of the Lemma requires a strong con-dition: independence among phase noise samples Θ(ti). Butthis independence condition is not unrealistic, as phase noisesamples can be uncorrelated and Gaussian distributed, andthus independent. This can be the case, e.g., with phasenoise generated by cross-phase modulation caused by neigh-boring channels in multichannel fiber-optic communicationsystems [2], [16].IV. SUFFICIENT STATISTICThe average mutual information, per unit time, between theinput and the output of the continuous-time channel in (5) isdefined as [18, pag. 370]I(X;Y ) , limM→∞12M + 1limN→∞I(XN,M ;Y N,M ), (21)whereXN,M = (XM0 , XM1 , . . . , XMN ), (22)andXMn = (Xn,−M , Xn,−M+1, . . . , XnM ), (23)whereXnm = 〈X(t), φnm(t)〉 (24)=∑kAk 〈gk(t), φnm(t)〉 = Am n = 0,0 n > 0. (25)In other words, XN,M is a vector that collects the(N + 1) · (2M + 1) projections of process X ontothe basis functions φnm(t) for n = 0, 1, . . . , N andm = −M,−M + 1, . . . ,M . An analogous notation holds forprocess Y .The projection of Y onto φnm(t) isYnm = 〈Y (t), φnm(t)〉 (26)=〈X(t)ejΘ(t) +W (t), φnm(t)〉(27)=∑kAk〈gk(t)ejΘ(t), φnm(t)〉+Wnm (28)a.s.= µΘ∑kAk∫ ∞−∞gk(t)φ?nm(t) dt+Wnm (29)where the Wnm’s are iid circularly-symmetric Gaussian vari-ates, and the last step follows by the Lemma in the previoussection. By using (9) in (29) we haveYnma.s.= µΘAm +Wnm n = 0,Wnm n > 0. (30)Since Ynm does not carry information about the input Amfor n > 0, and the sequences of random variables convergealmost surely to their limits, we haveI(X;Y ) = limM→∞12M + 1I(XM0 ;YM0 ) (31)= limM→∞12M + 1M∑m=−MI(X0m;Y0m) (32)= I(X00;Y00) = I(A0;Y00) (33)where (32) follows because the channel is memoryless by (30),and (33) follows by stationarity and by (25). The result (33)shows that a sufficient statistic for inferring Ak given Y isY0k = 〈Y (t), φ0k(t)〉 = 〈Y (t), gk(t)〉 (34)that is the output of the sampled filter that is matched to g(t).V. DISCUSSIONBy virtue of (30), at the output of the sampled matched filterthe continuous-time white phase noise channel is equivalentto a discrete-time AWGN channelY0k = µΘAk +W0k (35)so the effect of the spectral loss µΘ is to reduce the SNR. Inthe limit of µΘ → 0, reliable communication over this channelis not possible.We conclude that the presence of rapidly varyingcontinuous-time phase noise affects not only the informationencoded in the phase of X(t), but also the informationencoded in its amplitude. We further conclude that even anoncoherent receiver is affected by white phase noise andsuffers from an SNR loss. This is true also for fiber-opticcommunication systems based on direct detection, e.g., whenthe front-end is a photodiode that performs photon counting,the numbers of photons being proportional to the energy thephotodiode harvests. The output of the photodiode is usuallymodeled as the modulus-square of the field at the input, soin principle the phase noise would not be an issue. But realphotodiodes are bandlimited and should be modeled as beingpreceded by a bandpass filter that reduces the energy collectedby the receiver, and the presence of these filters will causeSNR loss.VI. CONCLUSIONWe have discussed the modeling of the continuous-timeAWGN channel in the presence of white (memoryless) phasenoise. We have found that, under mild conditions, the sufficientstatistic for this channel is the output of a baud-sampledmatched filter, and that the channel is equivalent to a discrete-time AWGN channel with an SNR penalty.ACKNOWLEDGMENTL. Barletta was supported by Technische Universita¨tMu¨nchen - Institute for Advanced Study, funded by theGerman Excellence Initiative. G. Kramer was supported byan Alexander von Humboldt Professorship endowed by theGerman Federal Ministry of Education and Research.REFERENCES[1] G. 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On continuous-time white phase noise channels

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